Universality theorems for configuration spaces of planar linkages

Contemporary Mathematics, 2001

We survey and expand 1 on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into complex projective space (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when n = 1 is given by a non-trivial extension which we determine.

Canadian Mathematical Bulletin, 1999

We give a \wall-crossing" formula for computing the topology of the moduli space of a closed n-gon linkage on S 2 . We do this by determining the Morse theory of the function n on the moduli space of n-gon linkages which is given by the length of the last side{the length of the last side is allowed to vary, the rst (n ? 1) sidelengths are xed. We obtain a Morse function on the (n ? 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of n are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of n at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages. the property that 0 < r i < ; 1 i n. In this case P is determined by its vertices u 1 ; : : : ; u n and we may write P = u = (u 1 ; : : : ; u n ).

1b.is paper describes the generation of configuration space obstacles in the plane, arising from the motion of objects with algebraic curve boundaries, moving with fixed orientation amongst obstacles bounded by algebraic curves. We show that the boundary of the configuration space obstacles are the envelopes of algebraic ooundary curves of the reversed object, reversed with respect to a reference point of the object, with the reference point moving on the physical obstacle. Two different algebraic methods are given to generate the boundary of the configuration space obstacles, both of time complexity o(nlog2n) for degree n algebraic curves. The task of finding collision free motion is then relatively simple and a number of polynomial time approaches are described.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily C 1-small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of "special inscribed configurations" inside families of spheres embedded in R n using differential topology. For instance, there is a C 1dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a C 1-dense family of smooth embedded (n − 1)-spheres in R n where each sphere has a family of inscribed regular n-simplices with the homology of O(n).

Annals of Mathematics, 2003

We give infinite series of groups Γ and of compact complex surfaces of general type S with fundamental group Γ such that 1) Any surface S ′ with the same Euler number as S, and fundamental group Γ, is diffeomorphic to S. 2) The moduli space of S consists of exactly two connected components, exchanged by complex conjugation. * The research of the author was performed in the realm of the SCHWERPUNKT "Globale Methode in der komplexen Geometrie", and of the EAGER EEC Project.

Geometry and Topology of Caustics – Caustics '02, 2003

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

2007

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has n arms such that each arm is of length 1 + 1 and has a rotational joint in the middle, and that the endpoint of the k-th arm is fixed to Re 2(k−1)π n i. Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given.

Bulletin of The Australian Mathematical Society, 2007

A method of embedding nk configurations into projective space of k-1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a "complementary" n n-k "theorem" about projective space (over a field or skew-field F) from any n* theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues IO3 (also in 3d-space), Mobius 84 (in 3d-space), and the resulting 7t in 3d-space, 96 in 5d-space, and IO7 in 6d-space. (The Mobius configuration is self-complementary.) There are some n/t configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

arXiv: Algebraic Topology, 2017

Let $F(X,k)$ be the configuration spaces of ordered $k-$tuples of distinct points in the space $X$. Using the Fadell and Neuwirth&#39;s fibration, we prove that the configuration space $F(M,k)$ of certain topological manifolds $M$, is not contractible.

Advances in Mathematics, 2007